...hism (theorems 8.5, 9.4). This categorical account can guide development of operators and laws for an axiomatic presentation, augmenting the laws of first order imperative programming (Morgan, 1994b; =-=Hoare et al., 1987-=-). With axiomatics in mind, most of our results are proved by inequational reasoning from the defining properties of the power adjunction, lifted exponent, etc. The remaining results, e.g., lemma 7.1 ...

...tal correctness of imperative programs remain unsatisfactory for a number of reasons. (a) There are inherent limitations even for relative completeness of proof systems, even for partial correctness (=-=Cousot, 1990-=-). (b) There are semantic difficulties involving interference and interactions between local variables and higher order procedures. Fully abstract semantics has proved to be surprisingly elusive (O'He...

...asible programs, the semantics of specifications involves technical complications such as predicates that are not Scott-open sets. (It is perspicuous to view Scott-open sets as observable predicates (=-=Abramsky, 1991-=-), but an accurate theory of effectiveness requires more restricted predicates, and specifications require less restricted ones.) Methodologists have prefered models like binary relations or predicate...

...hroughout the paper. To make the paper more accessible, some definitions are less general than usual (Street, A Categorical Model for Higher Order Imperative Programming 9 1980; Makkai & Par'e, 1989; =-=Borceux, 1994-=-). When patience wears thin, you may wish to to skip to x3.2 and use the index as needed. The application of function g to argument a is denoted by ga, even in cases like Gra where the function has a ...

...rictions imposed on global variables for reasons of efficiency. The location-free predicate transformer semantics of local variables is nicely suited to such languages (Morgan, 1994b; Naumann, 1994a; =-=Naumann, 1996-=-). Turning to (c) and (d), there is no shortage of alternative models for first order imperative programs. Operational semantics is needed for low-level correctness proofs (e.g., for compilers). Domai...

...n thanks to restrictions imposed on global variables for reasons of efficiency. The location-free predicate transformer semantics of local variables is nicely suited to such languages (Morgan, 1994b; =-=Naumann, 1994-=-a; Naumann, 1996). Turning to (c) and (d), there is no shortage of alternative models for first order imperative programs. Operational semantics is needed for low-level correctness proofs (e.g., for c...

...he fixed rules of a calculus; mature engineering disciplines freely mix formal calculations with reasoning about a standard model. As for (b), I prefer to deal with interference-controlled languages (=-=O'Hearn, 1993-=-), at least until (c) and (d) are addressed adequately. Also, much of the difficulty is absent in languages like C and Oberon thanks to restrictions imposed on global variables for reasons of efficien...

...ducts and lax coproducts in Tran, which are quite similar to those in ITran. The lax coexponent in Tran is discussed in order to motivate the move to ITran. Except where noted, the results appear in (=-=Martin, 1991; Naumann,-=- 1995b). Proposition 4.4. Each coproduct A + B of sets is a categorical product in Tran, equipped with the Stone duals ['] of the injection functions ' as &quot;projections&quot;. The associated funct...

...ntic difficulties involving interference and interactions between local variables and higher order procedures. Fully abstract semantics has proved to be surprisingly elusive (O'Hearn & Tennant, 1995; =-=Sieber, 1995-=-). (c) Domain-theoretic semantics of imperative programs is given using secondary categorical structure (?-lifting, powerdomains) rather than being intrinsic to the semantic category. This complicates...

...ike C and Oberon thanks to restrictions imposed on global variables for reasons of efficiency. The location-free predicate transformer semantics of local variables is nicely suited to such languages (=-=Morgan, 1994-=-b; Naumann, 1994a; Naumann, 1996). Turning to (c) and (d), there is no shortage of alternative models for first order imperative programs. Operational semantics is needed for low-level correctness pro...

...that the full homset is not appropriate as a data type: Programs are strict, positively conjunctive, and continuous, so those morphisms are the only ones that would be stored or passed as parameters (=-=Naumann, 1995b). Taking-=- for A/B just those &quot;healthy&quot; morphisms makes coap healthy, but it does not give an adjunction in the subprocat of such transformers, and the problem with id / B remains. Moreover, for purpo...

...gory, not of the span category. For transformer semantics of higher order commands in refinement calculus the full homset is the requisite coexponent (Naumann, 1994a) but it satisfies very weak laws (=-=Naumann, 1995-=-b). My contribution here is to use skew spans starting from posets instead of sets. The resulting category ITran is comprised of transformers between lattices of upward closed sets. Because predicates...

... contrast, some notions of &quot;categories of partial maps&quot; use only equations, but many equations are needed to replace the usual ones (other categorical treatments of partiality make more use =-=of order (Fiore, 1995-=-)). Inequational weakenings of categorical notions are used by He and Hoare (1989) to classify language features and phenomena. There has been extensive work on calculi of transformers for programs in...